The homology of the groupoid of the self-similar dihedral group

TL;DR

This paper presents an example of a specific groupoid where the rational homology diverges from the K-theory of its associated C*-algebra, and confirms it satisfies Matui's AH-conjecture.

Contribution

It provides a novel example of a groupoid with unique homological properties and verifies the AH-conjecture for this case.

Findings

Rational homology differs from K-theory in the example

The example satisfies Matui's AH-conjecture

Highlights the complexity of groupoid homology and K-theory relationship

Abstract

We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.